7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Largeeddy simulation of the twophase flow in an unbaffled stirred tank
N. Lamarque* Y. Dolias! O. Lebaiguel M. Bertrand 5
CERFACS, 42 avenue Gaspard Coriolis 31057 Toulouse Cedex, France
t CEA, Nuclear Energy Division, Nuclear Technology Department, SE2T, LITA, F38054 Grenoble, Cedex 9, France
SCEA, Nuclear Energy Division, Reactor Studies Department, SSTH, LDAL, F38054 Grenoble, Cedex 9, France
SCEA, Nuclear Energy Division, RadioChemistry & Processes Department, SCPS, LMPR, F30207 BagnolssurCeze, France
olivier.lebaigue @cea.fr
Keywords: Largeeddy simulation, freesurface vortex, unbaffled stirred reactor, hydrodynamic instabilities
Abstract
Unbaffled tank are rarely used in chemical engineering devices because baffles enhance turbulent macromixing and
avoid the freesurface vortex formation in the case of a twophase flow. However, in some applications, the flow in
an unbaffled reactor can offer some advantages. This study describes the main characteristics of the unsteady velocity
field in such a configuration, with the help of Large Eddy Simulation. Results are compared with experimental mea
surements and theoretical developments, showing a good agreement. Hydrodynamic instabilities are also highlighted
and the freesurface vortex formation is described. Turbulent stresses and spectral analyses show the high complexity
of such a flow and the difficulty to model it.
Introduction
This work focuses on the description of the turbulent
twophase flow inside a cylindrical unbaffled reac
tor. Mixing tanks are commonly used in chemical
engineering applications (Perry and Chilton). Most
often, reactors are baffled, so as to enhance mixing
and avoid freesurface vortex creation. Nevertheless,
those obstacles also create some recirculation zones
that can be the location of unwanted phenomena such
as reactant or particle accumulation. This can strongly
decrease the device efficiency or even raise security
concerns in some cases like nuclear fuel reprocessing.
To avoid such situations, unbaffled tanks are used.
Nevertheless, studies on the turbulent flow generated
in such reactors are still rare, despite some recent
interesting numerical and experimental studies (Alcamo
et al. 2005; Haque et al. 2006; Assirelli et al. 2008;
Mahmud et al. 2009; Lamarque et al. 2010). The main
difficulty, when simulating flows in unbaffled tanks, is
to take into account for the freesurface deformation.
Some successful attempts have been made in the past
using unsteady RANS methods, along with the Volume
Of Fluid method to describe the freesurface vortex
formation. Here, the twophase turbulent flow is studied
with the help of the Large Eddy Simulation (LES)
technique. The interface is captured using a variant of
the FrontTracking method of Tryggvason et al. (2001).
Experimental measurements, based on Laser Doppler
Velocimetry (LDV), enable to draw comparisons and
validate the numerical simulation. Mean and fluctu
ating velocity fields and freesurface deformation are
provided. A study of the turbulent stress tensor shows
that turbulence is far from being isotropic throughout
the tank, with very different behaviours depending
on the location, which is impossible to model with
an eddyviscosity lI\lpllcsis, Finally, energy spectra
confirm the complexity high unsteadiness of such a flow.
Experimental setup
The reactor studied here is an unbaffled cylindrical glass
vessel with a diameter T and a height H (see Fig. 1 and
patent by Auchapt and Ferlay (1981)). Their ratio is:
H/T = 1.65. The tank is filled with water, initially
at rest and then agitated by a cylindrical magnetic rod,
lying at the bottom and which length is noted D. In this
study, D/T 0.47 and the impeller Reynolds number,
Rea ND2/v, is around 70 000, with N, the impeller
rotation speed and v the kinematic viscosity.
T
Gas
Liquid
LD
D
Figure 1: Sketch of the reaction tank.
The measurement system consists in a twocompoi
LDV. The flow is seeded with Nylon particles, the
ameter of which is 4/zm. For each measurement p
tion, acquisitions generally last about 700 impeller
stations, which ensures to obtain from 1 000 to 70
samples. Data repeatability has been checked by rep
ing the measurements several times. Depending wi
the intersection volume is placed, each acquisition
vides mean and RMS axial velocity and either tanger
or radial velocity.
Computational tools
The flow inside the tank
Simulation (LES).
is solved using LargeE
Vu = 0,
+ (uv)
1
V + vAu
P
 V T"9' + f,
lent
di
osi
ro
000
eat
lere
pro
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
Atomique (CEA). Eqs.(1) and (2) are here solved using
a Fractional Step method (Kim and Moin 1985). The
space discretization relies on a mixed finite element for
mulation (Fortin 2006). It can be seen as a generalization
of the Marker and Cell (MAC) method of Harlow and
Welch (1965) for unstructured meshes. The time march
ing method is the explicit firstorder Euler scheme. Con
vection terms are handled with an FCT method (Kuzmin
and Turek 2002). The resulting scheme has been shown
to be both robust and accurate. Its performance in the
context of LES has been described in (Ducros et al.
2010).
To simulate freesurface flows, the same pressure and
velocity fields are used for both fluids (air and liquid).
Density and viscosities are constant within each phase.
The freesurface is captured using a variant of the Front
Tracking method of Tryggvason et al. (2001), called
Discontinuous FrontTracking method (DFT) (Mathieu
et al. 2003). It is tracked by a moving Lagrangian
grid, independent of the Eulerian finite element mesh
described above. This approach has many advantages: it
describes the freesurface motion according to the flow
and provides a sharp reconstruction of the phase indi
cator and a stable application of surface tension forces
(Mathieu et al. 2003).
itial The fluid is initially at rest and the interface is flat (h
Hi). At the beginning of the simulation, the liquid starts
being stirred by the magnetic rod. Here, it is modelled
with the Immersed Boundary Condition technique (Fad
lun et al. 2000). Its geometry and motion are represented
ddy with the use of another Lagrangian mesh and its effects
on the flow are imposed through source terms in the mo
mentum equations. NS equations are then solved in the
(1) fixed frame of the tank and the Eulerian mesh, used for
both fluids, is also fixed. The computational domain is
discretized with a fixed homogeneous unstructured grid,
(2) only composed of tetrahedra. There are 120 000 nodes
and 650 000 cells of approximately the same volume.
where the definition of the residualstress tensor r9"8 is:
Tsys = U 199U U 0 U.
u is the filtered velocity vector, p the pressure and u
stands for the sum of the other different filtered source
terms (such as surface tension or gravity field). Filtering
is here implicit and the cutoff length is the grid char
acteristic size. The residualstress tensor is expressed
using the WALE (WallAdapting Local Eddyviscosity)
model (Nicoud and Ducros 1999).
Simulations are performed using the Trio_U code
(Calvin et al. 2002; Mathieu et al. 2003; wwwtrio
u.cea.fr) developed at the Commissariat a l'tnergie
Flow description
Mean flow velocity. As there is no baffle in the tank,
the flow is mainly circular. Besides, the tangential mean
velocity component is dominant in comparison with the
others. Fig. 2 shows the evolution of the tangential ve
locity with respect to the radius r* 2r/T, at z*
z/H, 0.58. As described by Nagata (1975), liquid
has mainly a solid body rotation motion at the reactor
centre. According to Rankine's description, this corre
sponds to a forced vortex, as the tangential velocity lin
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
early increases with the radius, which gives:
r < r : (Ue) = Or,
with 0 ~ 0.95 (27N) in the calculation, which is fairly
close to the theory. It should be added that 0 is actu
ally not constant: it is always smaller than 27N and
decreases a little, as z increases, which has also been
observed experimentally. Beyond the critical radius re,
tangential velocity is inversely proportional to the ra
dius, which is a free vortex zone (Fig. 2). This is seen
both experimentally and numerically and:
r > r : (uo) =K (r) ,
where K is a real constant. Here, n 0.63 and also
slightly varies with z. Even though the impeller studied
here is not classical, this result is very close to that of
Smit and Diring (1991) (n 0.6). Moreover, n is found
to be 0.7 experimentally. Furthermore, the computa
tional values of the critical radius r, are nearly the same
as those calculated by EsseyricEmile (1994) from ex
perimental results, using theories of Nagata (1975) and
Le Lan and Angelino (1972).
A 0.4
V
 0.2
0.0
Forced Free
roe ,re.
vortex ;
*
vortex
0.0 0.5
c r*
Figure 2: Mean nondimensionalized tangential veloc
ity magnitude at z* = 0.58. Dashed line: forced vortex.
Dotdashed line: free vortex. r* is the critical radius.
Fig. 3 shows that the mean radial motion is negligible al
most everywhere, except at the bottom (fluid is expulsed
by the impeller) and near the freesurface (centripetal
motion). Despite the absence of any baffle, water has an
axial motion. The fluid goes up near the walls, while it
slowly sinks near the axis. As a consequence, there is
a large toroidal recirculation zone in the tank, as it can
be seen on Fig. 3. One should also note the presence of
a small secondary toroidal recirculation zone in the cor
ner, at the bottom. Fig. 4 points out the helical paths of
the fluid particles in the tank.
Comparisons with experimental results. Figs. 510
II
'II
I!
I'l
I I
Figure 3: Particle paths generated from the mean veloc
ity projected in (Oxz) plane.
7">
/s
\\
Figure 4:
flow).
Particle paths in the tank (built with the mean
provide a comparison between simulations and exper
imental measurements for the height z* 0.35 and
Figs.1114 for a radius r* 0.61. Velocities are di
mensionalized using a reference velocity Vo = 7ND.
0.0
U.2 U.4 U.b U.8 1.i
Figure 5: Mean radial velocity at z*
LES, Blue dots: experiments.
0.35. Solid line:
There is globally a good agreement, which validates the
numerical simulations. Those results also confirm the
0.0!
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.2 0.4 0.6
T*
A
* 0.(
V
0.8 1.0
0.0 0.2 0.4 0.6
T*
Figure 6: Fluctuating radial velocity at z* = 0.35. Solid
line: LES, Blue dots: experiments.
0.5
0.4
0.3
0.2
0.1
0.0
0.2 0.4 0.6
T*
Figure 9: Mean axial velocity at z* = 0.35. Solid line:
LES, Blue dots: experiments.
A
 o0.1(
V
0.8 1.0
.0 0.2 0.4 0.6
r*
0.8 1.0
Figure 7: Mean tangential velocity at z* = 0.35. Solid
line: LES, Blue dots: experiments.
0.20
A
V
0.05
 .u U.i U.4 U.0
r*
u.$ I.u
Figure 8: Fluctuating tangential velocity at z*
Solid line: LES, Blue dots: experiments.
0.35.
dominance of the tangential motion and a very small
mean radial motion, except near the impeller, where the
fluid is ejected towards the wall. Besides, it is interesting
to see note that fluctuations are very intense for all com
ponents, especially at the bottom of the tank. Mean and
Figure 10: Fluctuating axial velocity at z*
Solid line: LES, Blue dots: experiments.
0.35.
fluctuating axial velocity are of the same order. More
over, radial fluctuations are much stronger than the mean
counterparts (with the very exception of the region close
to the impeller). This indicates a strong unsteadiness of
the flow. Additional comparisons, at different heights,
can be found in (Lamarque et al. 2010). They mainly
lead to the same conclusions.
Flow unsteadiness. This unsteady activity seems to be
mainly due to hydrodynamic instabilities appearing in
the tank. Fig. 15 shows an instantaneous velocity field.
As mentioned, it can be seen that the rod generates a
strong motion at the bottom. Furthermore, the vectors
draw circular patterns in the freevortex zone.
The use of Qcriterion Jeong and Hussain (1995), to de
tect vortices, enables to highlight the forced vortex on
the central axis, a strong activity near the impeller and
helical hydrodynamic structures (see Fig. 16). Those he
lical instabilities have a long lifetime (several rod rota
tions). Thus, those coherent structures may have a key
role in mixing and particle segregation (which can ap
0.20r
A
0O.lc
V
0.0!
0.8 1.0
u
1
x" '?
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
0.25[
0.2 0.4 0.6
Z*
0.8 1.0
Figure 11: Mean radial velocity at r*
line: LES, Blue dots: experiments.
Figure 12: Fluctuating radial velocity at r*
Solid line: LES, Blue dots: experiments.
0.61. Solid
Figure 14: Fluctuating axial velocity at r*
Solid line: LES, Blue dots: experiments.
0.61.
Min I
= Max
0.25
0.20
A 0.15
S0.10
V 0.05
0.00
0.2 0 05.6 0
0.1.0 0.2 0.4 0.6 0.8 1.
Z*
Figure 13: Mean axial velocity at r*
LES, Blue dots: experiments.
0
0.61. Solid line:
pear during precipitation reactions).
Figure 15: Instantaneous velocity vectors in the liquid
coloured by the velocity magnitude. The freesurface
has been emphasized (red line).
Freesurface
One of the most visible consequences, when removing
the baffles and having no flat lip at the top of a mixing
tank, is the formation of a freesurface vortex. This phe
nomenon is due to the pressure gradient generated by the
rotation motion. Fig. 17 shows the time evolution of this
vortex. At the beginning of the simulation, the surface is
flat and 10 to 15 rod rotations are needed before seeing
a deformation of the freesurface. The vortex growth
is then very fast before slowly stabilizing. Except the
vortex tip, which has a small processing motion and os
cillates, the surface is quite stable. Besides, as it can
be seen in Fig. 18, the surface profile is well retrieved
in the simulation. The agreement with classical correla
tions such as Nagata's is good. Vortex maximum depth
0.61.
...
^ 
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
,'/ Nagata r = 0.241
//  Nagata r =0.266
Simulation
0.5
Figure 18: Freesurface
between theory and LES.
Figure 16: Isosurface of Qcriterion showing the forced
vortex, the activity at the bottom and helical hydrody
namic instabilities
is obtained with a difference from 5% to 15% depend
ing on the choice of correlation for rc Lamarque et al.
(2010).
Min
 Max
vortex profiles. Comparison
tion, we give a similar study, taking advantage of the
numerical simulation.
Applying the classical Reynolds average to the NS equa
tions leads to the unclosed term Reynolds stress tensor
R, which stands for the mean turbulent stresses:
R = ((u < u >) (u < u >)).
We use the timeaveraged velocity fields provided by
the simulation to build the turbulent stress anisotropy
tensor. It can be analyzed using different graphic rep
resentations. A complete description can be found in
(Pope 2000; Escudi6 et al. 2004; Simonsen and Krogstad
2005). The method chosen here makes use of the
anisotropy tensor:
R 2
B = I.
k 3
k 1/2 Rkk is the turbulent kinetic energy. Th analysis
shows that the anisotropy tensor B eigenvalues1, noted
s and t lie in a realisability triangle. Indeed, due to the
definition of B and with the convention
Figure 17: Freesurface vortex evolution in time. Top:
instantaneous velocity magnitude. Bottom: Freesurface
coloured by curvature (black: minimum, white: maxi
mum).
Turbulence
Several past studies (Derksen et al. 1999; Galetti et al.
2004; Escudi6 et al. 2004; Hartmann et al. 2004) have
highlighted the anisotropic behaviour of the Reynolds
stress tensor, especially in the vicinity of the impeller. In
particular, Escudi6 and Lin6 (2006) have used the Lum
ley triangle to illustrate this point. In the following sec
s >t > (s+t),
we have:
2 4
3 3
2 2
<<3
3 3
Fig. 19 shows that the boundaries of this realisability do
main have a meaning. Moreover considering relations
(6) and (7), eigenvalues of B only lie in the grey trian
gle of Fig. 19. With some algebra, one can show that
(Escudi6 and Lin6 2006):
Isotropic turbulence lies at the centroid,
'As B is traceless, the third eigenvalue is (s + t).
0.8
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
* Medians indicate an axisymmetric turbulence. It
can be seen in turbulent channel when going away
from the walls (right side) or in mixing layers
(rather left side),
* Side midpoints represent the twocomponent limit
(2C) of turbulence, which is usually found when
+ 0,
* Triangle vertices stand for a onecomponent state
(1C).
Figure 20: Decomposition of the tank into six zones.
Zones 1,2 and 3 are defined with respect of r*, while
zones A and B are defined with respect of z*.
Figure 19: (s, t) representation. The grey zone rep
resents the same realisability domain as in the Lumley
triangle, with the convention: s > t > (s + t).
As LES solution holds most of the turbulent fluctuations
(those resolved by the grid), it is possible to build a good
estimate of Reynolds tensor. Therefore, for each con
trol volume in the liquid, we have access to good esti
mates of s and t that would be provided by experimental
measurements or a Direct Numerical Simulation (DNS).
Consequently, as Hartmann et al. (2004) have done, we
show the distribution of the eigenvalues calculated from
the present LES results. Their locations in the aforemen
tioned triangles are provided below.
To simplify the survey, the vessel is decomposed into
different zones, as in (Escudi6 et al. (2004)). The
first three domains are defined with respect of r* (see
Fig. 20):
Zone 1: r* E [0, (' '.]
This zone corresponds to the forced vortex, where
fluid has a quasisolid body rotation and mean tan
gential velocity evolves as r* (Fig. 21).
Zone 2: r* e]0.26, 0.75]
In this bigger domain, mean tangential velocity de
creases as (r*) It is the free vortex zone and
stops when approaching walls (Fig. 22).
05 00 05
s
10 1.5
Figure 21: (s, ) (right) triangles for zone 1 (forced
vortex zone).
Zone 3: r* e]0.75, 1]
This last part lies in the outer part of the tank. Mean
tangential velocity follows neither of the above
mentioned laws as viscous and wall effects are sen
sible (Fig. 23).
In the three zones (Fig. 2123), (s, t) triangles clearly
highlight the turbulence anisotropy, as no dots are
situated close to the origin. On the other hand, it is
quite difficult to point out a clear and unique trend
for each zone. Indeed, the three zones have a great
number of locations where B shows no particular
behaviour (neither isotropic nor axisymmetric stresses).
However, some remarks can be made when comparing
the three zones. Near the wall (Fig. 23), triangles
show that turbulence tends to be twocomponent, as
it is observed in a channel when y+ 0 (Krogstad
and Torbergsen (2000)). This behaviour is due to the
wallnormal fluctuation suppression. As expected, this
/
Gas
Liquid
z
SrF
I I
I I
I 1
I I
I I
I I
I I
I I
Z*

1 2 3
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1.C
0.5
05 0.0 0.5
5
Figure 24:
stirrer).
05 00 05
1.0 1.5
(s, t) (right) triangles for zone A (near the
(s, t) (right) triangles for zone 2 (free vortex
0 5 0.0 0.5 1.0 1 5
Figure 25:
the stirrer).
(s, t) (right) triangles for zone B (far from
0.5 0.0 0.5
S
10 15
Figure 23: (s, t) (right) triangles for zone 3 (close to
walls).
is not observed in the other two zones. It seems there
is also a trend towards axisymmetric turbulence in the
freevortex zone (Fig. 22), as in pipe flows, far from the
walls, which is not seen in zone 3: no dot is situated
really close to the axisymmetric lines (see Fig. 23).
Finally, it is interesting to see that anisotropy is more
marked in the tank outer parts. The first decomposition
has enabled to compare turbulent stresses in the free and
forced vortex zones and close to the walls, regardless of
height. A second set of zones, defined with respect of
z*, is used to give another insight of turbulence structure
in the tank. Owing to its location, zone B presents most
of the characteristics of zones 2 and 3, which means
a large number of different kinds of turbulent stresses,
far from isotropy. Globally, Figs. 24 and 25 show that
turbulence is closer to isotropy in zone A. This implies
that anisotropy does not decay when fluid goes up and
this result is different from what is obtained with baffled
tanks. This is not surprising as shear stresses in the tank
and the fluid rotation both reinforce anisotropy (Jacquin
et al. 1990; C (i.ss.ing 2000). Similar conclusions have
been obtained using Lumley triangles and have been
reported in (Lamarque et al. 2010).
Figure 22:
zone).
10 1.5
'_1%k'
ljh
qmmih
O O0=04
Z2 O R R
Zl 1 1 :: :: 
RO R1 R2 R3 R4
Figure 26: Locations of the spectral acquisitions. Ex
perimental and/or LEScomputationissued spectra were
obtained approximately at: z* 0.11, 0.21, 0.35, 0.58
and 0.71; r* 0.00, 0.18,0.44, 0.71 and 0.89. Angular
positions were 0, 90, 180 and 270 for experiments and
0, 45, 90 .... 315 for computations.
Spectral study
In coherence with the other nondimensional quantities
used in the previous sections, the frequencies are
converted to nondimensional ones with the impeller
rotation speed N used as a reference frequency. Spectra
are computed from temporal signals whose duration
exceeds 7000 rod rotations for experimental signals and
200 rod rotations for LES computations.
Spectral peaks due to the rod and its harmonics.
Starting with the spectral analysis, we first looked at the
most obvious signature: the signature of the rod motion.
Close to the impeller, the dominant frequency is of
course twice the impeller rotation rate (i.e., f* = 2).
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
1 E02 
,' i
"., .r Ii,
I E 07 0
0 1 1 10 100
Figure 27: Energy spectrum of the vertical velocity
fluctuations at a position z* 0.11 and r* 0.18
(Location 1 on Fig. 26). Violet line: experiments (mea
sured at 0, 90, 180 and 270 at the same z* and r*),
Orange lines: LES computations (measured at 0, 45, 90
.. 315 at the same z* and r*). The black straight
line is a k 5/3 slope. The 1024 frequencies range from
f* z 0.1 to f* z 100. Except for this black line, the
curves are unfitted data, without any multiplying factor.
Figure 27 shows a typical spectrum of the vertical
velocity fluctuations at a position z* = 0.11 and
r* 0.18, i.e. close above the rod, in the forced vortex
region. Several harmonics of the main frequency are
observed, especially in the LES computation, namely
f* =2, 4 and 6. The real experimental signal exhibits
more dumped harmonics, for the harmonic f* = 6 is
barely present in the real fluid. This figure also shows
that the dispersion of the 4 experimental lines versus
angular position in the tank is moderate, except for
very low frequencies. Similarly, the dispersion of the
8 curves in the LES computation is very limited, even
at low frequencies (A more ideal geometry? A more
accurate positioning?). The peak close to f* 1 in
the simulation results is an artefact of an overestimated
epitrochoidal motion superimposed to the rod rotation
to mimic an offaxis precession of the centre of gravity
of the rod. This motion has been observed for low
height of liquid in the tank and postulated to exist when
the tank is properly filled. The absence of such a f* 1
peak in the experimental results shows that the real
precession motion is more limited with nominal liquid
height.
Spectral analysis of the dissipation slope. Figure 27
shows a unsatisfactory agreement between the high fre
quency trends. Clearly, the simulation seems to be over
dissipating for the highest frequencies in several loca
tions in the tank. This is probably due to a lack of res
olution. The cutoff frequency in the calculation is a lit
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
tie too low and as a consequence, the nonlinear con
vection scheme too much stabilizes the flow in those re
gions. However, this behaviour does not seem to affect
the overall energy of the fluctuations: the agreement be
tween experimental and computational lower frequency
levels is fairly good. A finer mesh should be used to limit
the impact of the convection scheme and have the cutoff
frequency in the inertial regime in those locations.
Spectral analysis of the vortical structures. Fig
ure 28 corresponds to a location in the free vortex and
higher in the tank (z* = 0.35 and r* = 0.89). As exhib
ited on Figure 27, the 8 (resp. 4) curves coming out from
the LES simulation (resp. the experiment) are very close
to each other, with simply reports the average curve of
the 8 (resp. 4) different angles. As for the Figure 27,
the slope at high frequency is close to a k 5/3 slope for
experimental data, whereas dissipation is stronger in the
LES. A more interesting feature lies in the lower fre
quencies: The series of three peaks (from the simula
tion results at f* z 0.17, 0.27 and 0.35) corresponds
to another flow feature. From the computational results,
we may conjecture that these three peaks are the results
of the vortical structures that are shown on Figure 16
(and also at midheight on the slice given by Figure 15).
Looking in the detailed velocity field of Figure 15 pro
vides two estimate of the main peak: the period of ro
tation of a single vertical structure at midheight corre
sponds to approximately f* z 0.17; moreover the pe
riod of rotation of the free vortex at the typical radius
r* z 0.70 is also in the range of f* z 0.17 0.20. We
may then conclude that this series of three peaks may
be associated to the presence of these vortical helices.
It seems that it is difficult to show the existence of these
helices from the averaged and fluctuating velocity fields,
partly due to the fact that these helices are not as pure
modes as the one that are computed from LES. The long
duration signals used to gain fine experimental signals
may blur these structures in local data.
Vortex tip frequency Figure 29 corresponds to a point
located on the revolution axis (z* 0.58 and r* 0.00.
It shows a peak at very low frequency that is associated
to the slow oscillation of the vortex tip, i.e. the lowest
point of the free surface. These oscillations are probably
associated to a low mode of the tank, but with an equilib
rium free surface that results from the balance between
centrifugal forces and the gravity forces. The oscillat
ing mode has been observed both in experiments and in
computations. A simple model and an analytical esti
mate of the frequency is still to be found. It does not
seem to be a problem already solved by existing litera
ture.
I E06
1 E 0 7  
0001 001 01 1 10 100
Figure 28: Energy spectrum of the radial velocity fluc
tuations at a position z* 0.35 and r* = 0.89 (Loca
tion 3 on Fig. 26). Red line: experiments, Blue line:
LES computation. The black straight line is a k 5/3
slope. The 16384 frequencies range from f* z 0.005
to f* z 100. Except for this black line, the curves are
unfitted data, without any multiplying factor.
1E05
0001 001 01 1 10 100
Figure 29: Energy spectrum of the vertical and hori
zontal velocity fluctuations at a position z* 0.58 and
r* 0.00. Orange and red lines: experiments (resp.
vertical and horizontal), Light and dark blue lines: LES
computation (resp. vertical and horizontal). The 16384
frequencies range from f* z 0.005 to f* z 100.
Conclusions
The present is one of the first attempts to carry out a
Large Eddy Simulation of the turbulent twophase flow
in an unbaffled cylindrical tank. The numerical simu
lation confirms some previous observations reported in
the literature. The flow is complex and there are two
main mixing zones: a forced vortex located near the axis
and a free vortex around the first one. Because there is
no baffle to break the tangential motion, it is predomi
nant throughout the tank, with the exception of the zones
situated near the impeller and the freesurface, which
is described here with the help of the Front Tracking
formulation. A freesurface vortex is formed because
of the pressure gradient generated by this circular mo
tion. Some additional features have been highlighted.
The flow is also very unsteady and strong hydrodynamic
instabilities have been pointed out. Comparisons both
with LDV experimental results and theory show the va
lidity of the methodology used here. Besides, an anal
ysis of the turbulent stress behaviour has been provided
and show the anisotropy and complexity of turbulence
throughout the reactor. Finally, even though a spectral
study shows some expectable differences between expe
rience and simulation, the latter one retrieve the most
iniliik.lii frequencies.
References
R. Alcamo, G. Micale, F. Grisafi, A. Brucato, and
M. Ciofalo. Largeeddy simulation of turbulent flow in
an unbaffled stirred tank driven by a Rushton turbine.
Chemical Engineering Science, 60:23032316, 2005.
M. Assirelli, W. Bujalski, A. Eaglesham, and A.W.
Nienow. Macro and micromixing studies in an unbaf
fled vessel agitated by a Rushton turbine. Chemical En
gineering Science, 63:3546, 2008.
P. Auchapt, A. Ferlay, 1981. Appareil a effet vortex pour
la fabrication d'un procede. Brevet FR 1 556 996.
C. Calvin, O. Cueto, and P. Emonot. An objectoriented
approach to the design of fluid mechanics software.
ESAIM: Mathematical Modelling and Numerical Anal
ysis, 36(5):907921, 2002.
P. Chassaing. Turbulence en mecanique des fluides,
analyse du phinomdne en vue de sa modilisation a
l'usage de l'inginieur. Cepadubs Editions, 2000.
J.J. Derksen, M.S. Doelman, and H.E.A. Van den Akker.
Threedimensional LDA measurements in the impeller
region of a turbulently stirred tank. Experiments in Flu
ids, 27:522532, 1999.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
F. Ducros, U. Bieder, O. Cioni, T. Fortin, B. Fournier,
G. Fauchet, and P. Qu6m6rd. Verification and validation
considerations regarding the qualification of numerical
schemes for LES for dilution problems. Nuclear Engi
neering and Design, Article in Press, 2010.
R. Escudi6 and A. Lin6. Analysis of turbulence
anisotropy in a mixing tank. Chemical Engineering Sci
ence, 61:27712779, 2006.
R. Escudi6, D. Bouyer, and A. Lin6. Characteriza
tion of trailing vortices generated by a Rushton turbine.
A.I.Ch.E. Journal, 50(1):7586, 2004.
C. EsseyricEmile. Modilisation du fonctionnement
d'un pricipitateur a effet vortex. These de doctorate de
l'Institut National Polytechnique de Lorraine, 1994.
E. Fadlun, R. Verzicco, P Orlandi, and J. MohdYusof.
Combined immersedboundary/finitedifference meth
ods for threedimensional complex flow simulations.
Journal of Computational Physics, 161, 2000.
T. Fortin. Une methode Elements Finis a decomposition
L2 d'ordre tlevi motive par la simulation d'ecoulement
diphasique bas Mach. These de doctorate de l'Universit6
de Paris VI, 2006.
C. Galetti, E. Brunazzi, S. Pintus, A. Paglianti, and
M. Yianneskis. A study of Reynolds stresses, triple
products and turbulence states in a radially stirred tank
with 3D laser anemometry. Journal of Computational
Physics, 82(9):12141228, 2004.
J.N. Haque, T. Mahmud, and K.J. Roberts. Model
ing turbulent flows with freesurface in unbaffled agi
tated vessels. Industrial and Engineering Chemistry Re
search, 45:28812891, 2006.
F.H. Harlow and J.E. Welch. Numerical calculation
of timedependent viscous incompressible flow of fluid
with free surface. Physics of Fluids, 8(12):21822189,
1965.
H. Hartmann, J. J. Derksen, C. Montavo, J. Pearson, I. S.
Hamill, and H. E. A. Van den Akker. Assessment of
large eddy and RANS stirred tank simulations by means
of LDA. Chemical Engineering Science, 59:24192432,
2004.
L. Jacquin, O. Leuchter, C. Cambon, and J. Mathieu.
Homogeneous turbulence in the presence of rotation.
Journal of Fluid Mechanics, 220:152, 1990.
J. Jeong and F. Hussain. On the identification of a vortex.
Journal of Fluid Mechanics, 285:6994, 1995.
7th International Conference on Multiphase Flow,
ICMF 2010, Tampa, FL, May 30 June 4, 2010
J. Kim and P. Moin. Application of a fractional
step method to incompressible NavierStokes equa
tions. Journal of Computational Physics, 59(2):308
323, 1985. ISSN 00219991.
P.A. Krogstad and L. E. Torbergsen. Invariant analysis
of turbulent pipe flow. Flow, Turbulence and Combus
tion, 64:161181, 2000.
D. Kuzmin and S. Turek. Flux correction tools for finite
elements. Journal of Computational Physics, 175:525
558, 2002.
N. Lamarque, B. Zopp6, O. Lebaigue, Y. Dolias,
M. Bertrand, and F. Ducros. Largeeddy simulation of
the turbulent freesurface flow in an unbaffled stirred
tank reactor. Chemical Engineering Science, Article in
Press, 2010.
A. Le Lan and H. Angelino. Etude du vortex dans les
cuves agit6es. Chemical Engineering Science, 27(11):
19691978, 1972.
T. Mahmud, J. Haque, K. Roberts, D. Rhodes, and
D. Wilkinson. Measurements and modelling of free
surface turbulent flows induced by a magnetic stirrer in
an unbaffled stirred tank reactor. Chemical Engineering
Science, 64:41974209, 2009.
B. Mathieu, O. Lebaigue, and L. Tadrist. Dynamic con
tact line model applied to single bubble growth. In 41st
European TwoPhase Flow Group Meeting, Trondheim,
Norway, 2003.
S. Nagata. Mixing: Principle and Applications. Wiley,
1975.
F. Nicoud and F. Ducros. Subgridscale stress modelling
based on the square of the velocity gradient. Flow, Tur
bulence and Combustion, 62(3):183200, 1999.
R. Perry and C. Chilton. 5th edn. McGrawHill, Chemi
cal Engineers' Handbook, 1973.
S. B. Pope. Turbulent flows. Cambridge University
Press, 2000.
A. J. Simonsen and P.A. Krogstad. Turbulent stress in
variant analysis: Clarification of existing terminology.
Physics of Fluids, 17(8):0880103, 2005.
L. Smit and J. During. Vortex geometry in stirred ves
sels. In Proceedings of the 7th European Congress of
Mixing, pages 633639, Bruges, Belgium, 1991.
G. Tryggvason, B. Bunner, A. Esmaeeli, D. Juric, N. Al
Rawahi, W. Tauber, J. Han, S. Nas, and Y.J. Jan. A
fronttracking method for the computations of multi
phase flow. Journal of Computational Physics, 169:
708759,2001.
